Spatial decay of discretely self-similar solutions to the Navier-Stokes equations

Zachary Bradshaw, Patrick Phelps

Published 2022-02-16Version 1

Forward self-similar and discretely self-similar weak solutions of the Navier-Stokes equations are known to exist globally in time for large self-similar and discretely self-similar initial data and are known to be regular outside of a space-time paraboloid. In this paper, we establish spatial decay rates for such solutions which hold in the region of regularity provided the initial data has locally sub-critical regularity away from the origin. In particular, we (1) lower the H\"older regularity of the data required to obtain an optimal decay rate for the nonlinear part of the flow compared to the existing literature, (2) establish new decay rates without logarithmic corrections for some smooth data, (3) provide new decay rates for solutions with rough data, and, as an application of our decay rates, (4) provide new upper bounds on how rapidly potentially {non-unique}, scaling invariant local energy solutions can separate away from the origin.